Ch. 3: Auctions in Corporate Finance 95
value is a dominant strategy.^8 Staying in the auction beyond the point of the bid equaling
one’s value cannot be rational. Likewise, if the item is about to be won by someone else
at a bid less thanvi, then bidderishould be willing to bid a bit higher than the current
bid, for if such a bid wins the auction it will yield a profit.
An open auction therefore will quite easily find the second-highest valuation and es-
tablish that as the price—for bidding will cease once the bidder with the second-highest
valuation is no longer willing to bid more.^9 The expected price in the open auction
is therefore the expected value of the second-highest valuation, the same as for the
first-price sealed-bid auction. This result is an implication of the Revenue Equivalence
Theorem; we return to a more general statement of that below.
Turn now to a second-price sealed-bid auction: in such an auction, sealed bids are
submitted and rules call for the highest bid to win but that the price paid will be the
second-highest bid submitted. With these rules it is again a dominant strategy to sub-
mit a bid equal to one’s valuation. Bidding more than one’s value would mean possibly
winning at a price in excess of value. Bidding less than one’s value will mean possibly
forgoing an opportunity to buy the object at a price less than value. The key to under-
standing the second-price auction is to note that the linkage between one’s bid and the
price one pays has been severed; bidding equal to value to maximize the probability of
profitable wins becomes optimal. Thus, as is the case for the open auction, the second-
price sealed bid auction will also yield as a price the second-highest value out of theN
values held by the bidders.^10
All three auctions therefore yield the same expected price.^11 Note, however, that the
first-price and second-price (including the open auction as essentially a second-price
auction) auctions have equilibrium strategies that are easy to compute for both the mod-
eller and the bidder. Note also that the first-price auction gives a different (and less
volatile) price for any given set of bidders. It is also important that all three auctions
are efficient in that the bidder with the highest valuation is the winner. Auctions can be
seen as accomplishing two distinct tasks: reallocating ownership of an asset and deter-
mining a price for the transfer of ownership. Efficiency is an important characteristic of
any sales procedure, and auctions under private value assumptions should get the asset
(^8) A (weakly) dominant strategy in game theory is a strategy which does at least as well as any other strategy
no matter what strategies other agents use.
(^9) This neglects effects (usually unimportant) of a minimum bid increment.
(^10) Therefore, the second-price and the open (also called English or Ascending) auctions are “equivalent”
in the sense that they lead the bidders to bid or drop out at their private value for the object. However, this
“equivalence” holds only in the private values setting. If the other bidders’ signals or valuations are relevant
for a given bidder’s valuation of the object, this equivalence breaks down, as the open bids by the other bidders
conveys additional information.
(^11) The first-price auction is “strategically equivalent” to yet another auction known as the Dutch auction,
which an open descending price auction in which the auctioneer starts with a high price and then gradually
lowers the price until some bidder accepts the price. Provided that the object has not been sold yet, a bidder
will accept an asking price that equals her bid in the first-price auction. Strategic equivalence is a stronger
notion than the equivalence between the open and the second-price auctions.